Fourier series basic results

1. TARUN GEHLOT (B.E, CIVIL HONORS)
Fourier Series: Basic Results
Recall that the mathematical expression
is called a Fourier series.
Since this expression deals with convergence, we start by defining a similar expression
when the sum is finite.
Definition. A Fourier polynomial is an expression of the form
which may rewritten as
The constants a0, ai and bi, , are called the coefficients of Fn(x).
The Fourier polynomials are -periodic functions. Using the trigonometric identities
we can easily prove the integral formulas
(1)
for , we have

2. TARUN GEHLOT (B.E, CIVIL HONORS)
for n>0 we have
(2)
for m and n, we have
(3)
for , we have
(4)
for , we have
Using the above formulas, we can easily deduce the following result:
Theorem. Let
We have

3. TARUN GEHLOT (B.E, CIVIL HONORS)
This theorem helps associate a Fourier series to any -periodic function.
Definition. Let f(x) be a -periodic function which is integrableon . Set
The trigonometric series
is called the Fourier series associated to the function f(x). We will use the notation
Example. Find the Fourier series of the function
Answer. Since f(x) is odd, then an = 0, for . We turn our attention to the
coefficients bn. For any , we have
We deduce

4. TARUN GEHLOT (B.E, CIVIL HONORS)
Hence
Example. Find the Fourier series of the
Answer. We have
and
We obtain b2n = 0 and
TARUN GEHLOT (B.E, CIVIL HONORS)
Find the Fourier series of the function

5. TARUN GEHLOT (B.E, CIVIL HONORS)
Therefore, the Fourier series of
Example. Find the Fourier series of the function function
Answer. Since this function is the function of the example above minus the
So Therefore, the Fourier series of
TARUN GEHLOT (B.E, CIVIL HONORS)
Therefore, the Fourier series of f(x) is
Find the Fourier series of the function function
Since this function is the function of the example above minus the constant
So Therefore, the Fourier series of f(x) is
constant .

6. TARUN GEHLOT (B.E, CIVIL HONORS)
Remark. We defined the Fourier series for functions which are
wonder how to define a similar notion for functions which are
Assume that f(x) is defined and
The function F(x) is defined and integrableon
of F(x)
Using the substitution
Definition. Let f(x) be a function defined and integrable
of f(x) is
where
TARUN GEHLOT (B.E, CIVIL HONORS)
We defined the Fourier series for functions which are -periodic, one would
wonder how to define a similar notion for functions which are L-periodic.
) is defined and integrable on the interval [-L,L]. Set
) is defined and integrableon . Consider the Fourier series
, we obtain the following definition:
) be a function defined and integrable on [-L,L]. The Fourier series
periodic, one would
. Consider the Fourier series
]. The Fourier series

7. TARUN GEHLOT (B.E, CIVIL HONORS)
for .
Example. Find the Fourier series of
Answer. Since L = 2, we obtain
for . Therefore, we have

8. TARUN GEHLOT (B.E, CIVIL HONORS)TARUN GEHLOT (B.E, CIVIL HONORS)